DOCSLIB.ORG

Finite Resolutions of Modules for Reductive Algebraic Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finite Resolutions of Modules for Reductive Algebraic Groups View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA I@473488 (1986) Finite Resolutions of Modules for Reductive Algebraic Groups S. DONKIN School of Mathematical Sciences, Queen Mary College, Mile End Road, London El 4NS, England Communicafed by D. A. Buchsbaum Received January 22, 1985 INTRODUCTION A popular theme in recent work of Akin and Buchsbaum is the construc- tion of a finite left resolution of a suitable G&-module M by modules which are required to be direct sums of tensor products of exterior powers of the natural representation. In particular, in [Z, 31, for partitions 1 and p, resolutions are constructed for the modules L,(F) and L,,,(F) (the Schur functor corresponding to 1 and the skew Schur functor corresponding to (A, p), evaluated at F), where F is a free module over a field or the integers. The purpose of this paper is to characterise the GL,-modules which admit such a resolution as those modules which have a filtration in which each successivequotient has the form L,(F) for some partition p. By contrast with [2,3] we do not produce resolutions explicitly. Our methods are independent of those of Akin and Buchsbaum (we give a new proof of their result on the existence of a resolution for L,(F)) and are algebraic group theoretic in nature. We have therefore cast our main result (the theorem of Section 1) as a statement about resolutions for reductive algebraic groups over an algebraically closed field. One may frequently wish to work over a more general coefficient ring (see [l-3]) and so we treat in some detail Chevalley group schemes and general linear group schemes over a principal ideal domain (2.6 and 2.7), giving the appropriate versions of our main result and relating it to the work of Akin, Buchsbaum and Weyman. 473 0021-8693/86 $3.00 Copyright ?: 1986 by Academic Press, Inc. All rights of reproduclmn m any form reserved. 474 S. DoNKIN 1. RESOLUTIONSAND FILTRATIONS Let G be a reductive, connected, linear algebraic group over an algebraically closed field K. Let T be a maximal torus of G and let @ be the root system of G with respect to T. Let B denote the Bore1 subgroup of G containing T such that the non-zero weights of the adjoint action of T on the Lie algebra Lie(B) of B are the negative roots. Let X denote the charac- ter group of T. There is a natural partial order on X: we write j” < p if p - 3. is a sum of positive roots. All modules considered here for an algebraic group over K will be rational (see Section 1 of [ 131) and finite dimensional over K. For 1,E X we denote by K, the one dimensional B-module on which T acts with weight %. The set X+ of dominant weights consists of those k E X such that the induced module (see [ 111) Kj, 1g is non-zero and we write Y(n) for KL 1g. For I dominant Y(n) has a unique highest weight I which occurs with multiplicity one (see, for example, (1.4.3) of [ 141). A filtration of a G-module is declared to be good if each successivequotient is either 0 or isomorphic to Y(1) for some 1 E X+. We denote by 6 the class of G-modules which have a good filtration and by 6* the class of G- modules V such that the dual module V* = Hom,( V, K) belongs to 6. (If G is semisimple and simply connected then VE 8* means that V has a Weyl filtration; see [20, 5.23.) For VE 8 the multiplicity of Y(J) as a successive quotient in a good filtration is, by (12.1.l ) of [ 141, independent of which good filtration is chosen and we denote this multiplicity by (V: Y(n)). DEFINITION. A special resolving system is a subclass 6 of 6 n 6* which is closed under the formation of direct sums and tensor products and which contains, for each 1 E X+, some module CA such that Cj. has unique highest weight i and ,I occurs with multiplicity one. Let B denote a special resolving system. We shall need the following sim- ple fact. LEMMA 1. Zf 0 + V’ + V + V” + 0 is a short exact sequence of G- modules with V, V’ and V” in 8 and CE (5 then 0 -+ Horn&C, V’) -+ Hom,( C, V) + Hom,( C, V”) -+ 0 is exact. ProojI By the long exact sequence of cohomology it is enough to prove that the Hochschild cohomology group Ext&(C, V’) is 0. Now Extk(C, v’) rH’(G, C*@ v’) and, since C*, V’E 6, C* @IV’ has a filtration with successive quotients of the form Y(A)@ Y(p), I”, p E X+. RESOLUTIONS OF MODULES FOR REDUCTIVE GROUPS 475 However, it is a result of Cline, Parshall, Scott and van der Kallen ((3.3) Corollary of [ 133 for semisimple groups; see also (2.1.4) of [ 143 for reduc- tive groups) that H’(G, Y(n)@ Y(p)) =0 for all i>O. Hence H’(G, C* @ I”) = 0 and the proof is complete. Our main result is that a special resolving system is a resolving system for 8, or more precisely: THEOREM. A G-module M has a finite left 6 resolution if and only if ME6 If M has a finite left (5 resolution then there is an exact sequence o-+ v,-+ v+, + . + V, + V, + M + 0 with all Vi E (li. It follows from Proposition 3.2.4 of [14] and induction on m (see also 4.1 (2) of [6]) that M has a good filtration. To establish the reverse implication we use induction on the weights of M. For a G-module S we set o(S) = { ,u E X+: p is a weight of S}. The for- mal character of Y(n) (A E X’ ) is given by Weyl’s character formula (see, e.g., [4, (2.2.7)]) so it follows, from 21.3 Proposition of [7], that o( Y(A)) is saturated in the sense that whenever r E o( Y(n)), p E X+ and p < r, then ALEo( Y(n)). Thus w(N) is saturated for any G-module N with a good filtration. LEMMA 2. For any non-zero ME 6 there exists a short exact sequence of G-modules O+D’-+D-+M+O with DE 6, D’ E 6 and such that o(D’) is a proper subset of o(M). The Theorem follows now from Lemma 2 by induction on lo(M)1 by combining a resolution 0 + C, -+ C, _ i -+ . + C,, + D’ + 0 with the short exact sequence 0 -+ D’ + D + M+O to form a resolution o+c,+ ... +C,,+D-*M+O ofM. Proof of Lemma 2. We prove Lemma 2 by induction on lo(M Assume the result to be true for all non-zero NE@ with lo(N)1 < lo(M Let 1 be a highest weight of M. By Lemma 3.2.3 of [14] there is a sub- module M’ of M such that M’ E 8, M/M’ is a direct sum of copies of Y(n) (say n of them) and 1$ u(M’). Also by Lemma 3.2.3 of [ 141 there is a sub- module, say CA, of Cl such that C; E 8,1$ o(C;) and C,/C, is isomorphic to Y(1). Thus we may choose an epimorphism rc: C= @:= 1 CA+ M/M with kernel C’ = @;= 1 C”. By Lemma 1, there is a morphism cp:C + M such that qcp= rc, where q: M+ M/M’ is the quotient map. Let $: C@ M’ + M be the sum of cpand inclusion. Then II/ is surjective and the kernel consists of those pairs (c, m’) E CO M’ with q(c) + m’ = 0, i.e., the 476 S. DONKIN pairs (c, --q(c)) with cp(c)~M’. Now q(c) belongs to M’ if and only if qcp(c)= 0, i.e., if and only if X(C) = 0, that is, if and only if c E c’. Hence the kernel of $, say E, is {(c, -q(c)): c E Cl}, which is isomorphic to C’. We have o(M’)_cw(M) and i $u(M’) so that Jw(M’)) < Iw(M)I. If M’ = 0 we may take D = C, D’ = C’ in the statement of Lemma 2. Thus we may assume that M’ # 0 and so, by the inductive hypothesis, there is an epimorphism 5: Q -+ M’, for some Q E 6 with kernel Q’ E 6 satisfying u(Q’) c o(M’). Let [: D = C@ Q + C@ M’ be the sum of the identity map and < and let a: D + M be the composite t+G0 5. We must show that the kernel D’ of a belongs to 6 and that u(D’) is a proper subset of w(M). We have D’ = Ker a = Ker(rj 0 i) = [-‘(Ker $) = [ ~ ‘E. The kernel of the restriction of {: D’ -+ E is the kernel of [, i.e., Q’, and the image is E. Both Q’ and E (isomorphic to C’) have a good filtration and so D’ E 8. Also, we have w(D’) = w(Q’) u o(E) = co(Q’) u w(C’) c o(M) u w(C’) since E= c’ and w(Q’) &w(M’).
Load more

Recommended publications
  • Arxiv:2003.06292V1 [Math.GR] 12 Mar 2020 Eggnrtr N Ignlmti.Tedaoa Arxi Matrix Diagonal the Matrix
    ALGORITHMS IN LINEAR ALGEBRAIC GROUPS SUSHIL BHUNIA, AYAN MAHALANOBIS, PRALHAD SHINDE, AND ANUPAM SINGH ABSTRACT. This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decompositionwith respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups. 1. INTRODUCTION Spinor norm was first defined by Dieudonné and Kneser using Clifford algebras. Wall [21] defined the spinor norm using bilinear forms. These days, to compute the spinor norm, one uses the definition of Wall. In this paper, we develop a new definition of the spinor norm for split and twisted orthogonal groups. Our definition of the spinornorm is rich in the sense, that itis algorithmic in nature. Now one can compute spinor norm using a Gaussian elimination algorithm that we develop in this paper. This paper can be seen as an extension of our earlier work in the book chapter [3], where we described Gaussian elimination algorithms for orthogonal and symplectic groups in the context of public key cryptography. In computational group theory, one always looks for algorithms to solve the word problem. For a group G defined by a set of generators hXi = G, the problem is to write g ∈ G as a word in X: we say that this is the word problem for G (for details, see [18, Section 1.4]). Brooksbank [4] and Costi [10] developed algorithms similar to ours for classical groups over finite fields.
    [Show full text]
  • Unitary Group - Wikipedia
    Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1.
    [Show full text]
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
    [Show full text]
  • Lectures on Lie Groups Over Local Fields
    Lectures on Lie Groups over Local Fields Helge Gl¨ockner1 Abstract The goal of these notes is to provide an introduction to p-adic Lie groups and Lie groups over fields of Laurent series, with an emphasis on the dynamics of automorphisms and the specialization of Willis’ theory to this setting. In particular, we shall discuss the scale, tidy subgroups and contraction groups for automorphisms of Lie groups over local fields. Special attention is paid to the case of Lie groups over local fields of positive characteristic. Classification: Primary 22E20; Secondary 22D05, 22E35, 26E30, 37D10. Key words: Lie group, local field, ultrametric field, totally disconnected group, locally profinite group, Willis theory, tidy subgroup, contraction group, scale, Levi factor, invariant manifold, stable manifold, positive characteristic. Introduction Lie groups over local fields (notably p-adic Lie groups) are among those totally disconnected, locally compact groups which are both well accessible and arise frequently. For example, p-adic Lie groups play an important role in the theory of pro-p-groups (i.e., projective limits of finite p-groups), where they are called “analytic pro-p-groups.” In ground-breaking work in the 1960s, Michel Lazard obtained deep insights into the structure of analytic pro-p-groups and characterized them within the class of pro-p-groups [32] arXiv:0804.2234v5 [math.GR] 28 Dec 2016 (see [10] and [11] for later developments). It is possible to study Lie groups over local fields from various points of view and on various levels, taking more and more structure into account. At the most basic level, they can be considered as mere topological groups.
    [Show full text]
  • A Linear Algebraic Group? Skip Garibaldi
    WHATIS... ? a Linear Algebraic Group? Skip Garibaldi From a marketing perspective, algebraic groups group over F, a Lie group, or a topological group is are poorly named.They are not the groups you met a “group object” in the category of affine varieties as a student in abstract algebra, which I will call over F, smooth manifolds, or topological spaces, concrete groups for clarity. Rather, an algebraic respectively. For algebraic groups, this implies group is the analogue in algebra of a topological that the set G(K) is a concrete group for each field group (fromtopology) or a Liegroup (from analysis K containing F. and geometry). Algebraic groups provide a unifying language Examples for apparently different results in algebra and The basic example is the general linear group number theory. This unification can not only sim- GLn for which GLn(K) is the concrete group of plify proofs, it can also suggest generalizations invertible n-by-n matrices with entries in K. It is the and bring new tools to bear, such as Galois co- collection of solutions (t, X)—with t ∈ K and X an homology, Steenrod operations in Chow theory, n-by-n matrix over K—to the polynomial equation etc. t ·det X = 1. Similar reasoning shows that familiar matrix groups such as SLn, orthogonal groups, Definitions and symplectic groups can be viewed as linear A linear algebraic group over a field F is a smooth algebraic groups. The main difference here is that affine variety over F that is also a group, much instead of viewing them as collections of matrices like a topological group is a topological space over F, we view them as collections of matrices that is also a group and a Lie group is a smooth over every extension K of F.
    [Show full text]
  • Linear Algebraic Groups
    Clay Mathematics Proceedings Volume 4, 2005 Linear Algebraic Groups Fiona Murnaghan Abstract. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 1. Algebraic groups Let K be an algebraically closed field. An algebraic K-group G is an algebraic variety over K, and a group, such that the maps µ : G × G → G, µ(x, y) = xy, and ι : G → G, ι(x)= x−1, are morphisms of algebraic varieties. For convenience, in these notes, we will fix K and refer to an algebraic K-group as an algebraic group. If the variety G is affine, that is, G is an algebraic set (a Zariski-closed set) in Kn for some natural number n, we say that G is a linear algebraic group. If G and G′ are algebraic groups, a map ϕ : G → G′ is a homomorphism of algebraic groups if ϕ is a morphism of varieties and a group homomorphism. Similarly, ϕ is an isomorphism of algebraic groups if ϕ is an isomorphism of varieties and a group isomorphism. A closed subgroup of an algebraic group is an algebraic group. If H is a closed subgroup of a linear algebraic group G, then G/H can be made into a quasi- projective variety (a variety which is a locally closed subset of some projective space). If H is normal in G, then G/H (with the usual group structure) is a linear algebraic group. Let ϕ : G → G′ be a homomorphism of algebraic groups. Then the kernel of ϕ is a closed subgroup of G and the image of ϕ is a closed subgroup of G.
    [Show full text]
  • Arxiv:2101.09811V1 [Math.NT]
    RECENT DEVELOPMENTS IN THE THEORY OF LINEAR ALGEBRAIC GROUPS: GOOD REDUCTION AND FINITENESS PROPERTIES ANDREI S. RAPINCHUK AND IGOR A. RAPINCHUK The focus of this article is on the notion of good reduction for algebraic varieties and particularly for algebraic groups. Among the most influential results in this direction is the Finiteness Theorem of G. Faltings, which confirmed Shafarevich’s conjecture that, over a given number field, there exist only finitely many isomorphism classes of abelian varieties of a given dimension that have good reduction at all valuations of the field lying outside a fixed finite set. Until recently, however, similar questions have not been systematically considered in the context of linear algebraic groups. Our goal is to discuss the notion of good reduction for reductive linear algebraic groups and then formulate a finiteness conjecture concerning forms with good reduction over arbitrary finitely generated fields. This conjecture has already been confirmed in a number of cases, including for algebraic tori over all fields of characteristic zero and also for some semi-simple groups, but the general case remains open. What makes this conjecture even more interesting are its multiple connections with other finiteness properties of algebraic groups, first and foremost, with the conjectural properness of the global-to-local map in Galois cohomology that arises in the investigation of the Hasse principle. Moreover, it turns out that techniques based on the consideration of good reduction have applications far beyond the theory of algebraic groups: as an example, we will discuss a finiteness result arising in the analysis of length-commensurable Riemann surfaces that relies heavily on this approach.
    [Show full text]
  • Brauer Groups of Linear Algebraic Groups
    Volume 71, Number 2, September 1978 BRAUERGROUPS OF LINEARALGEBRAIC GROUPS WITH CHARACTERS ANDY R. MAGID Abstract. Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d - l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, characterless, linear algebraic group over an algebraically closed field of characteristic zero: the Brauer group is finite-in fact, it is the Schur multiplier of the fundamental group of the algebraic group [6, Theorem 4.1, p. 299]. In this note, we extend these calculations to an arbitrary connected linear group in characteristic zero. The main result is the determination of the Brauer group of a ¿/-dimen- sional affine algebraic torus, which is shown to be (Q/Z)(n) where n = d(d — l)/2. (This result is noted in [6, 4.8, p. 301] when d = 2.) We then show that if G is a connected linear algebraic group whose character group has Z-rank d, then the Brauer group of G is C X (Q/Z)<n) where n is as above and C is finite. We adopt the following notational conventions: 7ms an algebraically closed field of characteristic zero, and T = (F*)id) is a ^-dimensional affine torus over F.
    [Show full text]
  • Linear Algebraic Groups
    Linear algebraic groups N. Perrin November 9, 2015 2 Contents 1 First definitions and properties 7 1.1 Algebraic groups . .7 1.1.1 Definitions . .7 1.1.2 Chevalley's Theorem . .7 1.1.3 Hopf algebras . .8 1.1.4 Examples . .8 1.2 First properties . 10 1.2.1 Connected components . 10 1.2.2 Image of a group homomorphism . 10 1.2.3 Subgroup generated by subvarieties . 11 1.3 Action on a variety . 12 1.3.1 Definition . 12 1.3.2 First properties . 12 1.3.3 Affine algebraic groups are linear . 14 2 Tangent spaces and Lie algebras 15 2.1 Derivations and tangent spaces . 15 2.1.1 Derivations . 15 2.1.2 Tangent spaces . 16 2.1.3 Distributions . 18 2.2 Lie algebra of an algebraic group . 18 2.2.1 Lie algebra . 18 2.2.2 Invariant derivations . 19 2.2.3 The distribution algebra . 20 2.2.4 Envelopping algebra . 22 2.2.5 Examples . 22 2.3 Derived action on a representation . 23 2.3.1 Derived action . 23 2.3.2 Stabilisor of the ideal of a closed subgroup . 24 2.3.3 Adjoint actions . 25 3 Semisimple and unipotent elements 29 3.1 Jordan decomposition . 29 3.1.1 Jordan decomposition in GL(V ).......................... 29 3.1.2 Jordan decomposition in G ............................. 30 3.2 Semisimple, unipotent and nilpotent elements . 31 3.3 Commutative groups . 32 3.3.1 Diagonalisable groups . 32 3 4 CONTENTS 3.3.2 Structure of commutative groups . 33 4 Diagonalisable groups and Tori 35 4.1 Structure theorem for diagonalisable groups .
    [Show full text]
  • (Linear) Algebraic Groups 1 Basic Definitions and Main Examples
    Study Group: (Linear) Algebraic Groups 1 Basic Denitions and Main Examples (Matt) Denition 1.1. Let (where is some eld) then n is an ane algebraic I C K[x] K VI = fP 2 A : f(P ) = 08f 2 Ig set. If I is prime, then VI is an ane algebraic variety. Denition 1.2. A linear algebraic group, G, is a variety V=K with a group structure such that the group operations are morphisms and V is ane. K-rational point e 2 G and K-morphisms µ : G × G ! G (µ(x; y) = xy) and i : G ! G (i(x) = x−1) K[G] is the K algebra of regular functions in V . ∆ : K[G] ! K[G] ⊗K K[G] the comultiplication, ι : K[G] ! K[G] is the coinverse We get the following axioms: (∆ ⊗ 1) ◦ ∆ = (1 ⊗ ∆) ◦ ∆ Example. : where , 1. , ∼ . n2 ∼ . Ga µ(x; y) = x+y a; y 2 A K[x] ∆ : x 7! x⊗1+1⊗y 2 K[x]⊗K[y] = K[x; y] Ga = Mn(K) : ∗ 2 dened by . Then . So Gm K ! V = (x; y) 2 A jxy = 1 t 7! (t; 1=t) (x1; y1) · (x2; y2) = (x1x2; y1y2) ∆ : x 7! x ⊗ x; y 7! y ⊗ y. Note that =∼ GL (K) p p Gm p 1 p p p Let , , −1 x1−y1 2 . Let K = Q( 2)=Q (x1 + y1 2)(x2 + y2 2) = (x1x2 + 2y1y2) + (x1y2 + x2y1) 2 (x1 + y1 2) = 2 2 x1−2y1 2 2, then we can use . , . g = x1−2y1 V (g 6= 0) ∆ : x1 7! x1⊗x2+2y1⊗y2 y1 7! x1⊗y2+x2⊗y1 ι : x1 7! x1=g; y1 7! −y1=g n .
    [Show full text]
  • EXPLICIT REALIZATION of the DICKSON GROUPS G2(Q) AS GALOIS GROUPS
    Pacific Journal of Mathematics EXPLICIT REALIZATION OF THE DICKSON GROUPS G2(q) AS GALOIS GROUPS Gunter Malle Volume 212 No. 1 November 2003 PACIFIC JOURNAL OF MATHEMATICS Vol. 212, No. 1, 2003 EXPLICIT REALIZATION OF THE DICKSON GROUPS G2(q) AS GALOIS GROUPS Gunter Malle For any prime power q we determine a polynomial fq(X) ∈ Fq(t, u)[X] whose Galois group over Fq(t, u) is the Dickson group G2(q). The construction uses a criterion and a method due to Matzat. 1. Introduction. In this paper we are concerned with the construction of polynomials whose Galois groups are the exceptional simple Chevalley groups G2(q), q a prime power, first discovered by Dickson; see Theorems 4.1 and 4.3. It was shown by Nori [7] that all semisimple simply-connected linear alge- braic groups over Fq do occur as Galois groups of regular extension of regular function fields over Fq, but his proof does not give an explicit equation or even a constructive method for obtaining such extensions. On the other hand, in a long series of papers Abhyankar has given families of polynomials for groups of classical types (see [1] and the references cited there). His ad hoc approach hasn’t yet led to families with groups of exceptional type (but see [2] for a different construction of polynomials with Galois group the simple groups of Suzuki). Thus it seems natural to try to fill this gap. In his recent paper Matzat [6] describes an algorithmic approach which reduces the construction of generating polynomials for such extensions to certain group theoretic calculations.
    [Show full text]
  • Math 224 - Representations of Reductive Lie Groups
    Math 224 - Representations of Reductive Lie Groups Taught by Dennis Gaitsgory Notes by Dongryul Kim Spring 2018 This was a course taught by Dennis Gaitsgory. The class met on Tuesdays and Thursdays from 11:30am to 1pm, and the main textbook was Springer's Linear Algebraic Groups. There were a few assignments near the beginning of the course, and a final paper. Contents 1 January 23, 2018 4 1.1 Linear algebraic groups . .4 1.2 Representations of linear algebraic groups . .6 2 January 25, 2018 7 2.1 Embedding of linear algebraic groups . .7 2.2 The regular representation . .8 3 February 1, 2018 10 3.1 Examples of groups and classification of representations . 10 3.2 Tannaka duality . 11 4 February 6, 2018 14 4.1 Connected component . 14 4.2 Image and surjectivity . 15 5 February 8, 2018 17 5.1 Fun with Frobenius . 17 5.2 More on orbits . 17 5.3 Jordan decomposition . 18 6 February 13, 2018 20 6.1 Representations of unipotent groups . 20 6.2 Groups of dimension 1 . 22 1 Last Update: August 27, 2018 7 February 15, 2018 23 7.1 Commutative groups with semi-simple points . 23 8 February 20, 2018 26 8.1 Continuous family of automorphisms . 26 8.2 Automorphisms of projective space . 28 9 February 22, 2018 29 9.1 Ind-schemes . 29 10 February 27, 2018 33 10.1 Comparing reduced parts of automorphism functors . 33 10.2 Lie algebras . 34 11 March 1, 2018 36 11.1 Equivariant quasicoherent sheaf . 36 11.2 Distributions .
    [Show full text]